A058481 -id:A058481 - cp's OEIS frontend

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A076217 a(1)=1, a(n) = a(n-1) + n * sign(n-a(n-1)).

Original entry on oeis.org

1, 3, 3, 7, 2, 8, 1, 9, 9, 19, 8, 20, 7, 21, 6, 22, 5, 23, 4, 24, 3, 25, 2, 26, 1, 27, 27, 55, 26, 56, 25, 57, 24, 58, 23, 59, 22, 60, 21, 61, 20, 62, 19, 63, 18, 64, 17, 65, 16, 66, 15, 67, 14, 68, 13, 69, 12, 70, 11, 71, 10, 72, 9, 73, 8, 74, 7, 75, 6, 76, 5, 77, 4, 78, 3, 79, 2, 80

Offset: 1

Benoit Cloitre, Nov 03 2002

a(n) = 1 correspond to n = A058481(m). - Bill McEachen, Aug 31 2023

			a(2) = a(1)+sign(2-a(1))*2 = 1 + 2 = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005132.

Cf. A057427, A058481.

a076217 n = a076217_list !! (n-1)
a076217_list = 1 : zipWith (+) a076217_list
   (zipWith (*) [2..] $ map a057427 $ zipWith (-) [2..] a076217_list)
-- Reinhard Zumkeller, Apr 21 2013

RecurrenceTable[{a[1]==1,a[n]==a[n-1]+n Sign[n-a[n-1]]},a[n],{n,80}] (* Harvey P. Dale, Jun 14 2011 *)

alist(N) = my(r, t=0); vector(N, i, t=r=t+i*sign(i-t)); \\ Ruud H.G. van Tol, May 10 2024

If 3^n>2*m>= 2*3^(n-1); a(3^n-2*m) = m; if 3^n>2*m+1>=2*3^(n-1)+1 a(3^n-2*m-1) = 3^n - m; special case of partial sum: sum(k=1, 3^n, a(k)) = (3/8)*(9^n-1) + (3^(n+1)-1)/2.
Conjecture: a(n) = -a(n-1)+a(n-2)+a(n-3) for n>5.  G.f.: -x*(27*x^28 +54*x^27 +27*x^26 +9*x^10 +18*x^9 +9*x^8 +3*x^4 +6*x^3 +5*x^2 +4*x +1) / ((x -1)*(x +1)^2). - Colin Barker, Feb 25 2013
Regarding Barker's conjectured recurrence, it seems to fail at n= powers of 3, and the 2 successive terms. So it holds except for n= 9-11, 27-29, 81-83, 243-245, .... - Bill McEachen, Mar 21 2025

A112027 a(1)=1; then successively add 1, divide by 2, add 2 and then total up the last 4 terms.

Original entry on oeis.org

1, 2, 1, 3, 7, 8, 4, 6, 25, 26, 13, 15, 79, 80, 40, 42, 241, 242, 121, 123, 727, 728, 364, 366, 2185, 2186, 1093, 1095, 6559, 6560, 3280, 3282, 19681, 19682, 9841, 9843, 59047, 59048, 29524, 29526, 177145, 177146, 88573, 88575, 531439, 531440, 265720, 265722, 1594321

Offset: 1

N. J. A. Sloane, Nov 24 2005

Joshua Zucker, Posting to Seq Fan mailing list, Nov 24 2005

Paolo Xausa, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-3).

Quadrusections: A058481, A024023, A003462, A067771.

a[1]:=1; k:=1; for n from 1 to 16 do k:=k+1; a[k]:=a[k-1]+1; k:=k+1; a[k]:=a[k-1]/2; k:=k+1; a[k]:=a[k-1]+2; k:=k+1; a[k]:=a[k-1]+a[k-2]+a[k-3]+a[k-4]; od;

LinearRecurrence[{0, 0, 0, 4, 0, 0, 0, -3}, {1, 2, 1, 3, 7, 8, 4, 6}, 50] (* Paolo Xausa, May 20 2024 *)

G.f.: -x*(6*x^7-3*x^4-3*x^3-x^2-2*x-1) / ((x-1)*(x+1)*(x^2+1)*(3*x^4-1)). - Colin Barker, Jul 28 2013

Definition found by Franklin T. Adams-Watters, Feb 01 2006

More terms from N. J. A. Sloane, Feb 22 2006

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).

Original entry on oeis.org

0, 1, 9, 52, 283, 1561, 8930, 53411, 334785, 2199034, 15119621, 108644581, 814474176, 6358910949, 51615342685, 434865155292, 3796991928727, 34308796490005, 320379418256794, 3087939032182127, 30683582797977749, 313977721545709002, 3305220440084030809, 35759627532783842561

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

nonn

Stirling transform of the cubes (A000578).

Exponential convolution of the sequences A000110 and A058481 (with a(0) = 0).

			E.g.f.: A(x) = x/1! + 9*x^2/2! + 52*x^3/3! + 283*x^4/4! + 1561*x^5/5! + 8930*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..572
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000110, A000578, A005494, A033452, A058481, A186021.

b:= proc(n, m) option remember; `if`(n=0,
       m^3, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 15 2022

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] - 1] (Exp[3 x] - 2 Exp[x] + 1), {x, 0, 23}], x]
Table[Sum[StirlingS2[n, k] k^3, {k, 0, n}], {n, 0, 23}]
Table[Sum[Binomial[n, k] BellB[n-k] (3^k - 2), {k, 1, n}], {n, 0, 23}]
Table[BellB[n+3] - 3*BellB[n+2] + BellB[n], {n, 0, 23}] (* Vaclav Kotesovec, Aug 06 2021 *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000578(k).

a(n) = A000110(n) + A005494(n) - A186021(n+1).

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 29, 9, 1, 1, 11, 41, 79, 61, 11, 1, 1, 13, 61, 169, 241, 125, 13, 1, 1, 15, 85, 311, 681, 727, 253, 15, 1, 1, 17, 113, 517, 1561, 2729, 2185, 509, 17, 1, 1, 19, 145, 799, 3109, 7811, 10921, 6559, 1021, 19, 1

Offset: 0

Philippe Deléham, Feb 24 2014

			Square array begins:
1..1...1.....1......1.......1........1........1...
1..3...5.....7......9......11.......13.......15...
1..5..13....29.....61.....125......253......509...
1..7..25....79....241.....727.....2185.....6559...
1..9..41...169....681....2729....10921....43689...
1.11..61...311...1561....7811....39061...195311...
1.13..85...517...3109...18661...111973...671845...
1.15.113...799...5601...39215...274513..1921599...
1.17.145..1169...9361...74897...599185..4793489...
1.19.181..1639..14761..132859..1195741.10761679...
1.21.221..2221..22221..222221..2222221.22222221...

Cf. A238303.

T:= proc(n, k); if n=1 then 2*k+1 else (2*n^(k+1)-n-1)/(n-1) fi end:
seq(seq(T(n-k, k), k=0..n), n=0..10); # Georg Fischer, Oct 14 2023

T(0,k) = A000012(k) = 1;
T(1,k) = A005408(k) = 2k+1;
T(2,k) = A036563(k+2);
T(3,k) = A058481(k+1);
T(4,k) = A083584(k);
T(5,k) = A137410(k);
T(6,k) = A233325(k);
T(7,k) = A233326(k);
T(8,k) = A233328(k);
T(9,k) = A211866(k+1);
T(10,k) = A165402(k+1);
T(n,0) = A000012(n) = 1;
T(n,1) = A005408(n) = 2*n+1;
T(n,2) = A001844(n) = 2*n^2 + 2*n + 1.

Definition amended by Georg Fischer, Oct 14 2023

A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

Original entry on oeis.org

2, 4, 12, 18, 20, 28, 30, 31, 34, 35, 38, 44, 45, 49, 50, 58, 60, 75, 79, 97, 100, 103, 111, 113, 118, 120, 135, 141, 153, 154, 156, 166, 168, 171, 178, 181, 204, 219, 220, 239, 245, 247, 254, 260, 267, 269, 280, 286, 298, 307, 313

Offset: 1

Zak Seidov, Sep 25 2020

nonn

The corresponding values of Omega: 1, 1, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 2, 6, 5, 4, 3, 4, 4, 4, 2, 4, 3, 3, 7, 4, 2, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 6, 5, 6, 4, 5, 4, 5, 7, 6, 8.

			2 is a term since Omega(3^2 - 2) = Omega(7) = 1, and Omega(3^2 + 2) = Omega(11) = 1.

Cf. A001222, A014224, A051783, A058481, A168607.

Select[Range[200],PrimeOmega[3^#-2] == PrimeOmega[3^#+2]&]

for (k = 1, 200, if ((m = bigomega (3^k - 2)) == bigomega (3^k + 2), print (k ", " m ", ")))

a(36)-a(51) from Amiram Eldar, Sep 25 2020

cp's OEIS Frontend

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

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A076217 a(1)=1, a(n) = a(n-1) + n * sign(n-a(n-1)).

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A112027 a(1)=1; then successively add 1, divide by 2, add 2 and then total up the last 4 terms.

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

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A282179 E.g.f.: exp(exp(x) - 1)*(exp(3*x) - 2*exp(x) + 1).

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A238339 Square number array read by ascending antidiagonals: T(1,k) = 2*k + 1, and T(n,k) = (2*n^(k+1)-n-1)/(n-1) otherwise.

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A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

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A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A282179 E.g.f.: exp(exp(x) - 1)(exp(3x) - 2*exp(x) + 1).

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.